Sizu Riatto
Calculus · Chapter 15

Changing Together

When two quantities are tied by an equation, their rates are tied too. Write the equation that links them, then take the rate of both sides.

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Opens on Chapter 15’s example: Pilar’s clearwell, 40 feet across, filling at 1,500 gallons a minute; a 13-foot ladder whose foot, 5 feet out, slides at 2 feet a second; a puddle; a shadow.

1. A filling tank
2. A sliding ladder
3. A spreading circle
4. A shadow
Enter your numbers, then press Work it out.

How it works

In a straight-sided tank, volume = surface area × level, and the area never changes, so the level rises at inflow ÷ area (in cubic feet: there are about 7.48 gallons in a cubic foot). In a cone whose radius is k times its height, the volume is (1/3)π(kh)²h, so its rate with height is πk²h²: the wider it gets, the slower the level rises for the same inflow. For the ladder, x² + y² = L²; taking the rate of both sides gives 2x × (foot's speed) + 2y × (top's speed) = 0, so the top slides at x × (foot's speed) ÷ y, faster and faster as the top nears the ground.

From the book, Chapter 15. 1,500 gallons a minute is about 200.5 cubic feet; the clearwell’s surface is 1,256.6 square feet, so the level rises 0.16 feet (about 1.9 inches) a minute. The sand pile rises 0.12 feet a minute at 6 feet tall, 0.47 at 3. The ladder’s top is 12 feet up and slides at 0.83 feet a second. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.